Here, \(v\)  denotes the Killing vector field that generates the \(T^2\) -action and \(\kappa =g(v,\cdot )\)  is the one-form canonically associated to \(v\)  via the metric \(g\) . A detailed description of \(P^+_\omega \)  is provided in [1]} and here we only state the result,
\(P^+_\omega =\frac{1}{1+\cos ^2\omega }\left(1+\cos \omega \star -\sin ^2\omega \frac{\kappa \wedge \iota _v}{\Vert v\Vert ^2}\right),\) 
